For the past few weeks, my life has been devoted to the teaching of mathematics. I’ve been lecturing Calculus at the Summer School, and teaching school leavers 2 unit mathematics in the form of an intensive bridging course. I’ve had a chance to interact directly with over 100 students, and their attitudes towards mathematics were similar to those I’ve encountered while tutoring privately. I thought I’d blog about the difficulties many students face, and how I’ve attempted to tackle them.

It’s counterintuitive and (therefore) hard

Mathematics attempts to deal with some rather abstract concepts. Most people will know a bit of mental arithmetic, but even something as simple as the “times tables” are drilled into us at a young age as part of their mathematical training. People tend to remember some very well, but not others – the 6’s, 7’s and 8’s are particularly confusing for most, since their ‘pattern’ is not as easy to identify (even though the pattern they make is rarely emphasised – rote learning is the key, for some reason).

Asking students to add fractions is an order of magnitude more difficult – it makes sense (according to how they think about addition) that 1/2 + 1/3 = 2/5, rather than the correct answer of 5/6. The necessity of generating a ‘common denominator’ is not something students understand. The fact that 2/5 is less than 1/2, and that adding a third to a half shouldn’t make it less than what you started with, or what 2/5 represents, doesn’t occur to them. What’s worst is when they don’t even know whether they’re wrong or right. They just write stuff down and hope it turns out to be correct, using rules that are often completely inappropriate for the given situation. This stage, which I encounter frequently, is a classic case of a mathematical collapse. The student is left feeling helpless and stupid, and the more you feel like that when you’re engaged in any activity, the more likely you will be to avoid it in the future.

And that’s just fractions, something you should have mastered (according to the syllabus) by the end of year 7. Once we start talking about the trigonometric functions like sine and cosine, students can’t put all the pieces together. It’s a function, rather than a number – something students cannot have automatic intuition for. Functions are mathematical machines, invented in our brains. They don’t exist anywhere else but on our calculators (who have a computer-designed approximation anyway). Thus you cannot add sine or divide by sine without knowing what you’re doing – applying the rules designed for numbers aren’t going to work here.

What is sine? It’s a function of angle that gives the height of a triangle inscribed on a unit (radius 1) circle with the given angle. Here’s a nice .gif:
What sine does is not so hard to explain, and cosine is nicely symmetric – it gives the horizontal distance rather than the height of the same triangle. It’s apparently hard enough, though, that most teachers will resort to “It’s a button on your calculator that you push. It’s equal to Opposite over Hypotenuse”. People who get this explanation are receiving explanations on the order of “World War 2 happened because Nazis were evil”. Easy, but oh so wrong. Perhaps you are dumb enough to need that kind of oversimplified explanation, but I sincerely doubt it.

The tools that mathematicians have developed to understand things such as massive quantities, probability, algebra and calculus are deeply unintuitive. Every method is developed with particular objects in mind, and generalising to other objects is hard – if it works for one thing, why doesn’t it work all the time? These questions are hard to answer, and even harder to answer in a 40 minute period with 30 other children in the class. This leads me nicely into:

Mathematics relies on generalisations of previous knowledge

Whenever I start tutoring students, whether they’re in year 7 or year 12, I always start with fractions. I’ve never had a student, not once, who could perform all the (rather elementary) operations without any mistakes. When you ask students to deal with something like fractional indices, you’re assuming that they can work with fractions. How can a teacher even begin to talk about fractional indices while their students can’t add a half and a third?

Many students get a blank look on their face when they find out they’re wrong after attempting to apply their reason to a problem. They have no idea why they’re wrong, but they don’t know what they don’t know. They gaze off in a helpless stupor as you try to explain to them what they should have done. They’re sure it all makes sense, but they no longer have any idea of why one approach will give the answer, whilst another will not.

Mathematics relies on the foundations that were (or should have been) laid in previous years. If you miss out on those lessons in fractions, or spend an entire period thinking “I hate my teacher” rather than concentrating on the concept of negative numbers, you start to fall behind. Most students never recover to a point where they can enjoy maths again, because it will forever be an uphill battle.

It’s my challenge as a teacher to locate these misunderstandings, correct them and press on with the more difficult material before the stupor sets in.

Mathematics is poorly motivated

“When are we ever going to use this?” is a question I get asked all the time. Mathematics is the kind of subject where the motives aren’t always clear, but the question always seems like an excuse for not working or succeeding whenever my students ask it. Clearly, if it’s all just fanciful nothingness, it’s not so bad if I can’t do it, right?

Sure, students might not be able to see exactly how to use mathematics to improve their life, but I don’t see how it’s any different to how other subjects work. Any benefit you will get out of learning most subjects will be tangential. You never hear:

“This perfectly executed sentence got me my job. Thank you English!”
“Thank god I knew about the way castles were fortified against sieges – the zombies would have eaten me otherwise! Thank you History!”
…and so forth. In reality, it’s more:
“I’m a more critical thinker now, and have good analytical techniques when it comes to messages being communicated to me. Thank you, English/History!”
…which is not something you can quantify.

Mathematics is sometimes represented as learning to be like a calculator or a computer. Whilst mental computation is a useful skill, it’s certainly not what the subject is from about year 7 when we start talking pronumerals and make claims about all numbers. I think this is an attempt to quantify the skills you are learning, rather than appreciating the abstract and tangential benefits you will learn to master.

The response in recent years seems to have been to make mathematics very connected to the real world. Taking a look at the General Mathematics syllabus, you can see their attempts – there is a large portion devoted to financial mathematics, the most boring kind of maths. Seriously, depreciation models. Any interesting subjects (such as spherical geometry, probability and the normal distribution) are dumbed down to the point where virtually no mathematical understanding is required – simply some real-world flavour coating on easy substitution (put x into the equation in the formula sheet) problems. I can’t tell you how frustrated it makes me when I flick through the textbooks of my students and feel so uninspired. If I can’t feel excited about this stuff, how can they?

I think there’s a problem at the heart of mathematics that makes it all seem a little pointless, and it goes deeper than just the “when will I use it in the supermarket” question. Mathematics is all about deductive logic – making conclusions from a given set of premises:

In a sense, the conclusion was already hidden among the premises, that is, in order to communicate the three statements above, I only need to communicate the first two. A mathematical problem is therefore about discovering true things (that are very often profound) by looking at what premises and conclusions you have already reached or discovered, and about the economy of expression.

This is very different to inductive logic, which is what science is. It attempts to take what you know and generalise outwards from there, for example:

Premise: All the whimples I’ve ever met were awesome.
Conclusion: All whimples are awesome.

I find deductive reasoning all the more fascinating and wonderful, which is why I chose to become a mathematician (even though science is WAY AWESOME). However, I can see why many students might feel a sense of “why bother?” when it comes to mathematics. There is only one answer, it’s there, and someone else can figure it out, really. Or you can just copy it from Bob – the Back of the book – and it’s not really that big a deal. The answers have no real meaning in and of themselves, so what’s the harm in copying?

The great profundity that I find in mathematics is its ability to say such powerful things about such a general class of objects. By starting with a few premises, so much can be artfully deduced. Mathematics truly is an art of elegance, much like poetry – it’s all in the selection and expression of thought. How we learn mathematics is also similar to the way you learn to play a musical instrument, or even learn to dance – drills and scales and boring repetition are all designed to get you used to the tool (the instrument or your body) that you’ll be using for profound expression later on.

… but it’s hard to tell that to the girl who’s got to pass MATH1011 so she can go be a vet and never think about x’s and y’s ever, ever again. Whilst I think that mathematics is one of the most essential things in life, and that it teaches you about proof, how to be sure, how to listen and how to deal with so many problems you might face in life (be they logical conundrums or even moral quandaries), deals with the secrets of the universe and holds the most exquisite beauty possibly conceivable by the human mind… I can honestly (and grudgingly) say that you could do without formal training. In the same way that some people just don’t like music or reading or television or the internet and instead find something else to fill their time, I can see why you might just not enjoy anything about mathematics. I am confused by this response, but I can accept it. Give me 10 minutes, though, and I think I could tell you something mathematical that you’ll find quite awesome (I hope to do this for my next blog post!).

Mathematics is Inhuman

Mathematics is all about discipline. It teaches you to ignore your intuition on many levels, instead forcing you to use your reason in very specific ways in very specific contexts. It teaches you to keep all the facts and assumptions in mind when you’re talking about anything, as the most basic properties may hold the key to your success. It really does stretch the brain out to places humans really haven’t needed to go (in terms of evolution), so the fact that we can do it is truly amazing.

However, it is inhuman in the sense that there is never any grey areas. Your answer is either right or wrong – at least in all mathematics before third year pure. You cannot argue your way out of mathematics or ‘fudge’ it with natural talent the same way you can with many of the humanities – remember those times you didn’t read the book but still got a good mark on your book review? Or wrote an essay that was kinda crap but you made it sound impressive with your words? Not so in mathematics. If you don’t know SOHCAHTOA, and what it means, then there is no way you can get any marks in a trigenometry test. It’s not so much about the inherent difficulty of the subject, but more about what questions are asked. I try and encourage in my students an open, inquiring mind that forms its own mental models of what’s going on, their own mental methods for solving problems. I try and ask questions that have multiple answers, using open questions rather than “what is s when t=1”. By encouraging them to actively engage with it in a more organic way, they need less help in staying motivated.

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Those are some preliminary thoughts. I’ll work more on this stuff for more postings in the future, perhaps when I’m not feeling so sick and rundown. I sure do want to talk about general numeracy as well as the wonders of mathematics a whole bunch. Please comment and let me know of your own personal feelings about and encounters with mathematics, because I’m always, always interested!

In the meantime, I’d like to keep you abreast of the awesome internet phenomena, such as kittens inspired by kittens:
….that video has exploded across the internet in just a few days. Funny how a video like that can lie dormant for months, and then the right connections are made and the right sites hear about it and before you know it your video has over 2 million views.

Also, sorry about the late post. Next Sunday, we’ll have Finn delivering a post, and Aidan the Sunday after that.